By definition, if $\Phi$ and $\Phi'$ are root systems of the Euclidean spaces $E$ and $E'$, respectively, then an isomorphism $\Phi\to\Phi'$ is one that is induced by an isomorphism $E\to E'$ which ...

Let $\Phi$ be an irreducible root system and $\Phi^+$ be positive root system and $\Delta$ be base. For every positive root $\beta=\sum_{\alpha \in \Delta}m_\alpha\alpha$, the numbers $m_\alpha$ are ...

I have a question about Lie groups. Let $G$ be a simply connected semi-simple complex Lie group and $\mathfrak{g}$ its Lie algebra. We consider a Cartan-Weyl basis of $\mathfrak{g}$, giving the usual ...

Let $K$ be a compact, connected, simply connected Lie group with Lie algebra $\mathfrak k$ and Killing from $B_{\mathfrak k}$. It is well known that $B_{\mathfrak k}$ is a negative definite symmetric ...

sl(n) is the set of nxn matrices with trace=0.
i know that sl(n) is a finite dimensional simple lie algebra and the maximal toral subalgebra of a finite dimensional semi simple lie algebra is abelian.
...

I am reading a paper which begin by a reminder about root system associated to a simple lie algebra $\mathfrak g$. let $\mathfrak h\subset \mathfrak g$ a cartan subalgebra.
Question 1: It says that ...

Let $R$ be a root system with simple roots $\Delta$. For all $\alpha\in\Delta$ let $\omega_\alpha$ be the fundamental weight associated to $\alpha$. Is then the length of $\alpha$ equal to the length ...

The proposition is from Humphreys.
I don't understand how to prove the highlighted statements. How can I express a general element of K? I tried using Cartan decomposition of L but it doesn't work.
...

It is well-known that Petersen graph is an strongly regular graph with parameters (10,3,0,1) and can be considered as complement graph of $L(K_5)$ and its spectrum is $\{3,1^5,(-2)^4\}$.
Also, It is ...

As far as I understand, the Cartan matrix is associated with a unique semi simple algebra. How can we compute the norm of a root $\alpha$ from it since its components are invariant under rescaling?
...

On page 142, the textbook An Introduction to Lie Groups and Lie Algebras (by Kirillov) determines the fundamental group of the root system $A_2$.
Basically, the author says we have two simple roots ...

Consider a reflection group $W$ acting by isometries on a Euclidean space $V$. I want to understand the union of $(-1)$-eigenspaces for this action, the set
$$\{v \in V : \exists w \in W\ (w\cdot v = ...

Can someone explain what a "closure" of a Fundamental Weyl Chamber means? I assume it is related to an algebraic closure, but I don't see how. In addition, how does the Weyl group act on it and why ...

I know that the roots of a Lie Algebra are functionals such that if $\alpha$ is a root and $h \in \mathfrak h$ is an element of the Cartan subalgebra, then $\alpha(h)$ is an eigenvalue.
I'm looking ...

My question may seem silly at first, but currently I am not able to work out the question of finding all roots for the n-dimensional torus. At first, it seemed obvious to me that there are no roots at ...

I want to show that a coxeter graph $\Gamma$ is connected if and only if its root system $\Phi$ is irreducible. So let $\Delta$ be a simple system of $\Phi$, and $\Delta$ is also our simple system. ...

The goal is to give the Dynkin diagram of $SU(n)$. One can show that the complexification of the Lie algebra $\mathfrak{g}$ of $G$ is given by $\mathfrak{G}_{\mathbb{C}}=\mathfrak{sl}(n,\mathbb{C})$ ...

Let $S = \{v_1,\ldots,v_n\} \subset \mathbb{R}^n$ and let $T = \{w_1,\ldots,w_n\} \subset \mathbb{R}^n$ be such that the angle between $v_i$ and $v_j$ is equal to the angles between $w_i$ and $w_j$. ...

Let $L$ be a Lie algebra. For vector spaces, modules, Banach spaces, etc. we have the notion of a dual.
Question: Is it possible to define naturally a Lie algebra $L^*$ that is in some sense dual to ...

I have been given a project to describe the construction of the Lie algebra associated to a Kac-Moody root datum $D=(I,A,\Lambda, (c_i)_{i\in I}, (h_i)_{i\in I})$.
I only know basic definitions: that ...

Consider the root datum $(X^*, \Delta,X_*, \Delta^{\vee})$ of a reductive algebraic group, where $X^*$ is the lattice of characters of a maximal torus, $X_*$ the dual lattice (given by the 1-parameter ...